A pair \((A,B)\)  or, equivalently, a realization (REF )-() for which \(\begin{bmatrix}A-\lambda I_n & B\end{bmatrix}\)  has full row rank at \(\lambda \in \mathbb {C}\)  is called controllable at \(\lambda \) . Similarly, a pair \((C,A)\)  or realization (REF )-() that is observable at \(\lambda \)  is one for which the pair \((A^\top ,C^\top )\)  is deemed controllable at \(\lambda \) . Moreover, a controllable/observable matrix pair or realization (REF )-() is one that is controllable/observable at any \(\lambda \in \mathbb {C}\) . Furthermore, a realization whose order is the smallest out of all others for the same \(\mathbf {G}(\lambda )\)  is called minimal (see section 6.5.1 of [1]}). A realization (REF )-() is minimal if and only if it is controllable and observable.
